Answer. The proportion implies the ratio of a to b, but it does not state that ratio explicitly. What ratio has 3 to 4? 3 is three fourths of 4. Explicitly, then, that is the ratio of a to b. a is three fourths of b.

4 is four fifths of 5 (Lesson 2), but it is not obvious of what number 12 is four fifths.

Alternately, however, 4 is the third part of 12 -- or we could say that 4 has been multiplied by 3. Therefore, 5 also must be multiplied by 3 --

4 : 5 = 12 : 15

That is,

4 : 5 = 3 × 4 : 3 × 5.

This is called the theorem of the same multiple.

4 is four fifths of 5. But each 4 has that same ratio to each 5. Two 4's, then, upon adding them, will have that same ratio to two 5's. Three 4's will have that same ratio to three 5's. And so on. Any number of 4's will have that same ratio, four fifths, to an equal number of 5's.

Here is how we state the theorem:

If we multiply two numbers by the same number,then the products will have the same ratio as the numbers we multiplied.

(Euclid, VII. 17.)

Problem 6. Write five pairs of numbers that have the same ratio as 3 : 4.

Create them by taking the same multiple of both 3 and 4. For example,

6 : 8, 9 : 12, 12 : 16, 15 : 20, 18 : 24

Problem 7. Complete each proportion.

a) 4 : 9 = 8 : 18

b) 4 : 9 = 12 : 27

c) 4 : 9 = 16 : 36

d) 7 : 8 = 21 : 24

e) 9 : 5 = 63 : 35

f) 4 : 11 = 20 : 55

g) 2 : 9 = 16 : 72

h) 6 : 5 = 54 : 45

i) 8 : 3 = 56 : 21

Problem 9. Complete this proportion, 2.45 : 7 = 245 :
700.

Since 2.45 has been multiplied by 100, then 7 also must be multiplied by 100.

PQ is two fifths of RS. If PQ is 12 miles, then how long is RS?

Solution. Since PQ is two fifths of RS, then proportionally,

PQ : RS = 2 : 5.

If PQ is 12 miles, then

PQ : RS = 2 : 5 = 12 miles : ? miles.

That is, 12 miles corresponds to PQ and 2. And since 12 is 6 × 2, the missing term is 6 × 5:

PQ : RS = 2 : 5 = 12 miles : 30 miles.

RS is 30 miles.

Or, since

PQ : RS = 2 : 5,

then inversely,

RS : PQ = 5 : 2.

Now, what ratio has 5 to 2? 5 is two and a half times 2. RS therefore is two and a half times PQ. And if PQ is 12 miles, then RS is 24 + 6 = 30 miles.

When the terms of a ratio have no common divisors except 1, then we have expressed their ratio with the lowest terms. They are the smallest terms -- the smallest pair of numbers -- that have that ratio.

Problem 12. Explicitly, what ratio have the following? Express each ratio with the lowest terms.

This theorem, or at any rate its algebraic version, seems to be the only one taught in the schools, and it has become the mechanical method for solving all ratio problems. The student should resist that tempatation and should understand the facts of ratio and proportion. We include it here only for the purpose of explaining the following:

Example 3. If a and bare numbers such that four a's are equal to three b's,